Scattering of a spherical wave by a small ellipsoid

A point generated incident field impinges upon a small triaxial ellipsoid which is arbitrarily oriented with respect to the point source. The point source field is so modified as to be able to recover the corresponding results for plane wave incidence when the source recedes to infinity. The main difficulty in solving analytically this low-frequency scattering problem concerns the fitting of the spherical geometry, which characterizes the incident field, with the ellipsoidal geometry which is naturally adapted to the scatterer. A series of techniques has been used which lead finally to analytic solutions for the leading two low-frequency terms of the near as well as the far field. In contrast to the near-field approximations, which are expressed in terms of ellipsoidal eigenexpansions, the far field is furnished by a finite number of terms. This is very interesting because the constants entering the expressions of the Lame functions of degree higher than three are not obtainable analytically and therefore, in the near field, not even the Rayleigh approximation can be completely obtained. On the other hand, since only a few terms survive at the far field, the scattering amplitude and the scattering cross-section are derived in closed form. It is shown that, in practice, if the source is located a distance equal to five or six times the biggest semiaxis of the ellipsoid the Rayleigh term of the approximation behaves almost as the incident field was a plane wave. The special cases of spheroids, needles, discs, spheres as well as plane wave incidence are recovered. Finally, some theorems concerning monopole and dipole surface potentials are included.